# Calculating AUC (linear and log-linear)

When performing non-compartmental analysis, the area under the concentration-time curve (AUC) is calculated to determine the total drug exposure over a period of time. Together with Cmax, these two parameters are often used to define the systemic exposure of a drug for comparison purposes. For example, in bioequivalence trials, the entire statistical analysis is based on the comparison between formulations of AUC and Cmax. While the mathematics involved in the calculation of AUC are simple, there are nuances to the methods that are often misunderstood. Hopefully I can review some of the key details here. You can also view my video on YouTube.

Although AUC can be calculated directly from primary PK parameters (CL and V), I will discuss only the numerical estimation of AUC using noncompartmental analysis techniques in this blog post.

#### Linear Trapezoidal Method

The linear trapezoidal method uses linear interpolation between data points to calculate the AUC. This method is required by the OGD and FDA, and is the standard for bioequivalence trials. For a given time interval (t1 – t2), the AUC can be calculated as follows:

$AUC = \frac{1}{2}(C_1+C_2)(t_2-t_1)$

In essence the first two terms calculate the average concentration over the time interval. The last piece (t1 – t2) is the duration of time. So the linear method takes the average concentration (using linear methods) and applies it to the entire time interval. When you sum all of the intervals together, you will arrive at the total exposure from the first time point to the last. If you then divide the total AUC by the total time elapsed, you will arrive at the “average” concentration of drug in the body over the total time interval.

#### Logarithmic Trapezoidal Method

The logarithmic trapezoidal method uses logarithmic interpolation between data points to calculate the AUC. This method is more accurate when concentrations are decreasing because drug elimination is exponential (which makes it linear on a logarithmic scale). For a given time interval (t1 – t2), the AUC can be calculated as follows:

$AUC = \frac{C_1-C_2}{ln(C_1)-ln(C_2)}(t_2-t_1)$

This method assumes that C1 > C2. The fraction represents the logarithmic average of the two concentrations. Just as with the linear method, the average concentration is multiplied by the time interval.

#### Linear-Log Trapezoidal Method

This is a combination of the first two methods and is also called “linear-up log-down”. When concentrations are increasing (as in the absorption phase), the linear trapezoidal method is used. When concentrations are decreasing (as in the elimination phase), the logarithmic trapezoidal method is used. This method is thought to be the most “accurate” because the linear method is the best approximation of drug absorption while logarithmic decline is best modeled by the logarithmic trapezoidal method during drug elimination.

#### Why are there different methods?

The following figure demonstrates how the linear trapezoidal method overestimates the AUC during the elimination phase. The blue line represents mono-exponential decline of a drug. Samples were drawn at 16 and 20 hours. The red line represents the linear trapezoidal methods estimation of drug concentrations. As you can plainly see, the red line is higher than the blue line suggesting overestimation by the linear trapezoidal method.

Linear Trapezoidal Method

The logarithmic trapezoidal method accurately estimates mono-exponential decline of drug concentrations. However, during an absorption phase, the logarithmic trapezoidal method can underestimate the exposure.

I hope you have a better understanding of how to calculate AUC using the different methods that are available. And I hope you understand the basis of these methods and the pitfalls and limitations of each.

1. Hans says:

Any comment on the (typical?) practice of replacement of low concentration values below the limit of quantification (BLQ) by zero or BLQ/2 before such calculations?

• Great question Hans. Instead of answering in the comments, I will prepare a separate blog post on the topic. Thank you for such an engaging question.

2. balaj says:

Calculation of volume of distribution.
Vd is calulated by formula Vd = Dose/Cp0 & by another formula Vd = CL/Kel, but answer by these two method does not some similar, why & help to solve the problem to get the same answer by these formulas.

• Thank you for the question. Assuming you have an IV dose, and a quality estimate of Cp0, I believe the best method for calculating Vd is the first equation you presented (Vd = Dose/Cp0). The second equation should really be turned around differently and used as follows: Kel = CL/Vd. The parameter Kel is a secondary pharmacokinetic parameter and derived from the primary parameters CL and Vd. If you try to use it in the method you describe, any inaccuracies in Kel will be magnified in the estimate of Vd leading to the discrepancy that you noted. I hope that helps.

• balaji says:

Thank you Nathan for explanation, still one doubt remains is that, in WinNonlin NCA, Vd is calculated by Vd = CL/Kel & not by the other equation i.e. Vd=D/Cp0. When we put the value of Cp0 calculated after NCA analysis in this formula answer for Vd not matches with WinNonlin output value of Vd. Please clarify.

• WinNonlin is designed to calculate Vd using the clearance and terminal elimination rate constant. The reason these values do not match is due to the variability in the estimation of both Cp0 and kel. Since these two values are calculated independently from different parts of the plasma concentration-time curve, any variance from the “true” values will result in differences in Vd estimates. If you are using mean data, then these differences in Vd estimates may be magnified. I would recommend a nonlinear model-based fit to accurately determine the volume of distribution.

3. Hanna says:

There are some factors contributing to blood glucose level (BGL). How can I assess the influence of different doses and different insulin types (rapid and slow acting) while calculating AUC of BGL in type I diabetes patients that depend upon external insulin?

• Nathan Teuscher says:

4. JS says:

Can there be any situation where AUC(0-24) is greater than Cmax? I recently came across such data and want to be sure if that is a mistake.

• Nathan Teuscher says:

AUC(0-24) and Cmax are different parameters with different units. AUC(0-24) is a measure of exposure over time and has units of concentration*time (e.g. ng*h/mL). Cmax is a measure of peak exposure at a specific time point and has units of concentration (e.g. ng/mL). Therefore you cannot directly compare numbers across parameters. Perhaps you can share more information about the specific scenario so I can help answer the question.

-Nathan

5. UT says:

I have a IV 2-compt. PK profile. I am interested in calculating the distribution half-life ( alpha half-life) using non-compartmental analysis. For performing the computation, I am using the Phoenix Winnonlin software package. In the setup of non-compartmental module, I am selecting the distribution phase instead of the terminal elimination phase in the slope selector section.

I was wondering that by doing so am I violating any assumptions for non-compartmental analysis as we usually need to select the terminal phase for lamda z calculation?

• Nathan Teuscher says:

Simply selecting the timepoints during the “distribution phase” will not give an accurate representation of the distribution half-life. Simply selecting the early time points will provide an estimate of a rate constant that is a combination of the distribution and elimination processes. I would recommend performing curve stripping instead using Excel or some other spreadsheet program. You can refer to a previous post I made on this topic (link). You can also perform curve stripping with Phoenix WinNonlin by performing a model fit to the profile.

6. Tan says:

For toxicokinetic studies, we don’t always get all the time points needed to get a nice curve. In some cases, we may get 1 or 2 time points with measurable values, but the remaining part of the curve is BLQ. What is the minimum number of measurable concentration data needed to calculate AUC? What is the standard practice?

• Nathan Teuscher says:

I don’t know of a standard practice; however, my opinion is that at least 5 data points are necessary to calculate an AUC value. Although as the number of data points drops, the confidence should also drop that the AUC actually represents total exposure.

7. Kristina says:

Hi Nathan
I am interested in calculating AUC from 4 measurements; the first one before drug administration (on the second day after initiation of treatment), the following ones 1, 3 and 24 hours later. The medicine is administered once daily so the last measurement is taken right before the next dose is given. Cmax is reached approximately after one hour so I am thinking linear trapezoidal method for the first 2 measurements and logarithmic trapezoidal method for the rest. Does this make sence? I know 4 points are not a lot, but I am very interested in getting an AUC value.

• Nathan Teuscher says:

Kristina,

You can calculate an AUC from 4 concentration values; however, it may be highly variable. I would recommend using linear trapezoidal when concentrations are increasing and logarithmic when concentrations are decreasing. Assuming that drug increases from 0-1 hr and then decreases from 1-24 hr, then your suggested plan is appropriate.

Nathan

• Kristina says:

Thank you for your answer! I am measuring concentrations of Moxifloxacin, and my problem is that for some patients, the concentration is still increasing at 3 hours. For these patients I have increasing concentration from 0-3 hours and then of course decreasing concentration at 24 hours. I am thinking that Peak/MIC might be a better estimate to use for analysing data than AUC/MIA with such few measurements. What is your opinion on that?

Kristina

• Nathan Teuscher says:

The method you use to calculate AUC does not really affect the PK/PD relationships you are trying to make. I would actually recommend developing a population PK model to correlate with efficacy rather than noncompartmental analysis. There are many published articles on the population pharmacokinetics of moxifloxacin. Then you can use those concentrations to investigate pharmacodynamic responses.

• Kristina says:

With the risk of sounding stupid, can WinNonlin be used for developing a population PK model? I ask because I am currently looking into which program to use for PK/PD calculations.
I am interested in finding out whether peak concentration of Moxifloxacin 400 mg daily reaches >10 times the MIC for common bacteria causing community-acquired pneumonia (using MIC from EUCAST).

• Nathan Teuscher says:

Not a stupid question at all. Phoenix WinNonlin cannot be used to develop a population PK model. However, Phoenix NLME can be used. I write about the NLME package in another post that you can read here. The NLME package requires an additional license from Pharsight/Certara.

• Kristina says:

I find it hard to find guidelines on how to create a population PK model, like how many subjects to use. Is there a golden standard?

I´ve read what you wrote about Phoenix NLME. Would you recomend this program rather than NONMEM? And what about online training for a beginner like myself? Is there an online course for Phoenix NLME like there is for Phoenix WinNonlin?

• Nathan Teuscher says:

Kristina,
There is no “golden standard” for the number of subjects required to conduct a population PK analysis. The more data that you have, the more precise and accurate your model becomes. I use both Phoenix NLME and NONMEM. In my opinion, Phoenix NLME is easier for most people who are new to population PK anlaysis, and it appears to be just as powerful as NONMEM. NONMEM has a longer history, and can be more flexible in certain situations, but it requires more effort to learn how to use the software and related tools to help with data formatting and figure creation. I have prepared an online course Population PK for Beginners that provides an introduction to the methods and software used for analysis. I show how to use NONMEM and Phoenix NLME to perform basic population PK analysis.
Nathan